Question

If `f` is an even function defined on the interval `(-5,5),` then four real values of `x` satisfying the equation `f(x)=f((x+1)/(x+2))` are ______, __________, _______ and______.

Answer 1

Correct Answer - `((pm3 pm sqrt(5))/(2))`
Since, f is an even function,
then `f(-x)=f(x), AA x in (-5,5)`
Given, `f(x)=f((x+1)/(x+2)) " ...(i)" `
`rArr f(-x)=f((-x+1)/(-x+2))`
`rArr f(x)=f((-x+1)/(-x+2))" "[ because f(-x)=f(x)]`
Talking `f^(-1)` on both sides, we get
`x=(-x+1)/(-x+2)`
`rArr -x^(2)+2x= -x+1`
`rArr x^(2)-3x+1=0`
`rArr x=(3pm sqrt(9-4))/(2)=(3pm sqrt(5))/(2)`
Again, `f(x)=f((x+1)/(x+2))`
`rArr f(-x)=f((x+1)/(x+2)) " "[because f(-x)=f(x)]`
Taking `f^(-1)` on both sides, we get
`-x=(x+1)/(x+2)`
`rArr x^(2)+3x+1=0`
`rArr x=(-3 pm sqrt(9-4))/(2)=(-3pm sqrt(5))/(2)`
Therefore, four values of x are `(pm 3pm sqrt(5))/(2)`